low dimensional quantum gases
TRANSCRIPT
Low dimensionalquantum gases
T. Giamarchi
http://dqmp.unige.ch/giamarchi/
Plan of the lectures
More on TLL (factionalization, topology, etc.)Effect of perturbations (Lattice and disorder)
Notions of transport in 1DFrom 1D to 3D; Ladders; Dimensional crossover
Equilibrium, basic notions of Tomonaga-Luttinger liquid
Why one dimension ?
Three urban legends about 1D
It is a toy model to understand higher dimensional systems.
It does not exist in nature ! This is only for theorists !
Everything is understood there anyway !
One dimension is speciallyinteresting
• No individual excitation can exist (onlycollective ones)
• Strong quantum fluctuations
Difficult to order| | ie θψ ψ=
Many CM or cold atoms Systems
cb
Drastic evolution of the 1d world
New methods (DMRG, correlations from BA, etc.)
New systems (cold atoms, magnetic insulators, etc.)
New questions (strong SOC, out of equilibrium, etc)
How to treat ?
``Standard’’ many body theory
Exact Solutions (Bethe ansatz)
Field theories(bosonization, CFT)
Numerics(DMRG, MC, etc.)
ReferencesTG, Quantum physics in one dimension, Oxford (2004)
TG in ``Understanding Q. Phase Transitions’’, Ed. L. Carr, CRC Press (2010)
M. Cazalilla et al., Rev. Mod. Phys.83 1405 (2011)
TG, Int J. Mod. Phys. B 26 1244004 (2012)
TG, C. R. Acad. Sci. 17 322 (2016)
And now we start….
Typical problem (e.g. Bosons)
• Continuum:
• Lattice:
Luttinger liquid physics
Labelling the particles
1D: unique wayof labelling
φ(x) varies slowly
CDW
~ 0q 0~ 2q πρ
θ: superfluid phase
Quantum fluctuations
U0 1
K1 1
Correlations
1/2
S(q,! ) J.S. Caux et al PRA 74 031605 (2006)
Finite temperatureConformal theory
β
χ
K
r
21
βξβπ
/)2( xxK
ee −−=
Other 1D systems
Spins
Powerlaw correlation functions
Non universal exponents K(h,J)
Use boson or fermions mapping
( 1) cos(2 )i i iS e eθ θ φ+ = − +1 (-1) cos(2 )z iS φ φπ−
= ∇ +
Fermions
Right (+kF) and left (-kF) particles
Luttinger liquid concept
• How much is perturbative ?
• Nothing (Haldane):provided the correct u,K are used
• Low energy properties: Luttinger liquid(fermions, bosons, spins…)
Luttinger parameters
Attractive Hubbard model
TG + B. S. Shastry PRB 51 10915 (1995)
``Quantitative’’ theory possible
• Trick: use thermodynamics and BA or numerics
• Compressibility: u/K
• Response to a twist in boundary: u K
• Specific heat : T/u
• Etc.
Tonks limit
U = 1 : spinless fermions
Not for n(k) : ψF ≠ ψB
Free fermions: 21( ) (0) cos(2 )Fx k x
xρ ρ ⟨ ⟩ ∝
K=1
Note: 1/2
† 1( ) (0)B Bxx
ψ ψ ⟨ ⟩ ∝
Tests of Luttinger liquids
Organic conductors
c
b
A. Schwartz et al. PRB 58 1261 (1998)
First observation of LL/powerlaw !!
TG PRB (91) :Physica B 230 (1996)
Cold gases
Bosons (continuum)
Optical lattices (dilute)B. Paredes et al., Nature 429 277 (2004)
†( ) ( ) (0)ikxn k dxe xψ ψ= ⟨ ⟩∫
Atom chips
K large (42)
S. Hofferberth et al. Nat. Phys 4 489 (2008)
†
0( ) (0)
Ldr rψ ψ⟨ ⟩∫
Charge velocity
https://www.futurity.org/one-dimensional-electrons-physics-1858622/
Other important 1D propertiesFractionalization of excitations
1S∆ = − ( )E q=
1/ 2S∆ = − 1/ 2S∆ = −
Magnon
Spinons
1·i ii
H J S S += ∑
Fractionalization of excitations
Hidden (topological) order parameters
Continuum of excitations
E(k) = cos(k1) + cos(k2)
k = k1 + k2
Magnons and spinons: 1 = ½ + ½
Spin Charge
Spinon Holon
Deconstruction of the electron: spin-charge separation
arXiv:1605.05661v2
Doped Hubbard model
Spin
Spin-Charge Separation higher D ?
Charge
Energy increases with spin-charge separation
Confinement of spin-charge: « quasiparticle »
Confinement of spinons BaCoVOQ. Faure, S. Takayoshi, et al. Nat. Phys 14, 716 (2018)
Take home message
Good theoretical methods to deal withthe case of a ``simple’’ equilibrium 1d systems (analytics and numerics)
Stepping stone to go beyond: manyexciting questions and problems (out of equilibrium, disorder, many chains, etc.)
Controled experimental realizations in condensed matter and cold atoms
Effect of lattices:Mott transition
Mott transition
T
δ = n-1
Mott insulator (n=1)
T < T_N : antiferromagnetic phase
?
Periodic lattice
• Incommensurate: Q ≠ 2 π ρ0
• Commensutate: Q = 2 π ρ0
cos(2 ( ) )H dx x xφ δ= +∫
cos(2 ( ))H dx xφ= ∫
0 cos( ) ( )H dxV Qx xρ= ∫0(2 2 ( ))
0 0cos( ) i x xH dxV Qx e πρ φρ −= ∫
Competition
2 210 [ ( ( , )) ( ( , )) ]
2 xudxdS x u x
K ττ ϕ τ ϕ τ
π= ∂ + ∂∫
Beresinskii-Kosterlitz-Thoulesstransition at K=2
String orderparameter
0 0 cos(2 ( ))LS V dxd xρ τ ϕ= − ∫
BKT transition
BKT: remarkable transision going outside the paradigm of Landau’s phase transitions
A transition wihout an order parameter Topological Vortex excitations
, ,cos( )i j i j
i j i jH J S S J θ θ
⟨ ⟩ ⟨ ⟩
= − ⋅ = − −∑ ∑
Vortex operator
| |x+aiaPe x⟩ = ⟩ ( , ) ( , )x
x dx xθφ τ π τ−∞
′ ′= Π∫
x
τ
02π 1 1cos(2 ( , ))xφ τ
Vortex operator for θ
21 ( ) ( ) cos(2 )2 xKS dxd u g dx
u τθ θ φτπ
= ∂ + ∂ − ∫ ∫
K : inverse temperature
g : vortex fugacity
Mott insulator:φ is lockedDensity is fixed
Gap in the excitation spectrum
T. Kuhner et al. PRB 61 12474 (2000)
TG, Physica B 230 975(97):arXiv/0605472 (Salerno lectures);
Oxford (2004);
M. Cazalilla et al., Rev. Mod. Phys.83 1405 (2011)/( ) rG r e ξ−∝
E. Haller et al. Nature 466 597 (2010)
G. Boeris et al. PRA 93 93, 011601(R) (2016)
Shows:
K* = 2RenormalizedSine-Gordon
Non local (topological) order
( ) ~ ( )x xρ φ∇
Science (2011)
Topological excitations is the norm in 1D
Disorder (equilibrium)
Example: localization of 1D interacting bosons
2
1 1
1 ( ) ( )2 2
N Ni
i j ii i j i
PH V R R D Rm= ≠ =
= + − +∑ ∑ ∑
TG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)
Competition Disorder vs Interactions
Existence of a Many-Body localized phase: Bose glass
Bose glass phase
U [interactions]
Db
0
Uc
Bose GlassSuperfluid
† /( ) (0) ~ xx e ξψ ψ −⟨ ⟩
Other potentials: Biperiodics
G. Roux et al. PRA 78 023628 (2008);T. Roscilde, Phys. Rev. A 77, 063605 2008;X. Deng et al PRA 78, 013625 (2008);
Effect of interactions?Same as ``true’’ disorder ?
J. Vidal, D. Mouhanna, TG PRL 83 3908 (1999); PRB 65 014201 (2001)
0 0 1 1( ) cos( ) V cos( )V x V Q x Q x= +
• U= 0Aubry-André model
• Localization transition
Quasi-periodics and interactionsC. D’Errico, E. Lucioni et al. PRL (2014);L. Gori et al PRA 93 033650 (2016)
0 0 0 1( ) cos( ) V cos( )V x V Q x Q x= +
d.c. transport at finite T
• In 1D all states are exponentially localized
• Finite T: sweeps energy E with probability f(E)
• No interactions σ(T)=0 for all T !
• Needs coupling to a thermal bath (phonons, etc.)
• Or with interactions can the system be its own thermal bath ?
With a bath
d.c. transport (high T)TG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)
Transport with a Thermostat• Mott variable range hopping
• 1D with interactions T. Nattermann, TG, P. Le doussal, PRL 91, 056603 (2003)arXiv:cond-mat/0403487
Without a bath
• With interactions can other particles be a ``bath’’ for one particle ?
• Can the system reach the thermodynamic equilibrium ? And explore ergodically the phase space ?
Many body localizationBasko, Aleiner, Altshuler; Gornyi, Mirlin Polyakov; Huse, ……
Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, Rev. Mod. Phys. 91, 021001 (2019)
No thermostat σ(T) = 0 if T<T* even with interactions
The system is not ergodic even at finite temperatures/energies
Take home message
Remarkable interplay betweenlocalization and interactions
Consequences and challenges both in equilibrium (LIP) and out of equilibrium (MBL).
Experimental possibilities to explore these phenomenaNeed better tools !
Transport
Transport
Condensed matter: a «routine» probe for materials
Theoretically: complicated ! Out of equilibrium
Often (but not always !) : linear response I=G V
Typical situation
Questions
Kubo; Memory function; Landauer; Keldysh; …..
I=f(V) Reflects the properties of the system (interferences – metal, insulator, etc.)
Methods ?
Expectations for small V: contact system
V RIR R R== +
Other transport(s): spin, temperature, etc.
Cold atoms
1D ballistic: quantized conductance
Atomtronic
M. Lebrat, P. Grisins et al., PRX 8 011053 (18)
No interactions: band insulator
What happens with interactions?
Luther-Emery liquid
Gap in the spin sector (singlet pairing)
Conductance determined by the charge sector
Many-body insulator``pinned’’ L.E. liquid
Experimental evidence for L.E. liquid
Spin transport
A.-M. Visuri, M. Lebrat, S. Häusler, L.Corman and TG, PRR 2, 023062 (2020)
Spin transport and spin drag( )J j j Gσ σ µ µ↑ ↓ ↑ ↓= ⟨ − ⟩ = −
Spin Drag: µ↑∆ I↓ I µ↓ ↑= Γ∆
Solution (using Sine-Gordon)
tJ σφ∝ ∂
Comparison with experiments
Lindblad reservoirs and lossesT. Jin, M. Filippone, TG PRB 102 205131 (2021)
( , ) ( , )dxdt x t x tξ ρ∫
T. Jin, J. Ferreira, M. Filippone, TG arxiv/2106.14765
Ballistic ? Diffusive ?
Artificial gauge fields
Meissner effect in bosonicladders
E. Orignac, TG, PRB 64 144515 (2001)
A dl⋅ = Φ∫
Orbital currents (``Meissner’’ effect)
Field ``Hc1’’: appearance of vortices
Artificial gauge field (cold atoms)M Atala et al. Nat Phys, 10 588 (2014)
Hall effect
Non interacting ``simple’’
||
1h
VRI B n⊥= ∝
No interactions: curvature of fermi surface- topological formula (Thouless-Kohmoto)
With interactions: open question
Hall effect: measurements
Hall effect on laddersS. Greshner, M. Filippone, TG, PRL 122, 083402 (19)
Experiments: out of equilibrium
Analytic: calculation on a ring
Vanishing Hall responseM. Filippone, C.E. Bardyn, S. Greshner, TG, PRL 123, 086803 (19)
Exact zero of the Hall effect for LB geometry
Hall votage vs polarizationM. Buser, S. Greshner, U. Schollwöck, TG, PRL 126, 030501 (21)
Practical protocol to compute Hall voltage Vh from Hall polarization Ph
Dimensional crossover
• Largely open physics • Strong difficulties to treat
(analytics, numeric)
• Many studies limited to non-interacting case
• Allows to incorporate SOC and magnetic field effects
• Relevant for many experimental systems
c
b
D. Jaccard et al., J. Phys. C, 13 L89 (2001)
Deconfinement
TG Chemical Review 104 5037 (2004)
Deconfinement
0 5 10 15 20 25
6080
100
200
400
600 TMTTF2PF6
∆a
T*
ICSDW
CSDWSpin-Peierls
∆c
Activ
ation
ener
gy, T
* (K)
Pressure (kbar)
P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004)A. Pashkin, M. Dressel, M. Hanfland, C. A. Kuntscher, PRB 81 125109 (2010)
TG Chemical Review 104 5037 (2004)
Chain (Cluster) - DMFTS. Biermann, A. Georges, A. Lichtenstein, TG, PRL 87 276405 (2001)C. Berthod et al. PRL 97, 136401 (2006)
Bath must be self consistently determined
Transverse transport
Perpendicular hopping t’
T > t’: tunneling, not usual transport
2 1
11 14 2
( , ) ( , )( )T T
K K
ασ ω ω
α
−
−
∝
= + −
DMRG
Conclusions
Tour of one dimensional physics
Luttinger liquid theory provides a framework to study this physics, and to go beyond
Remarkable and complementary realization in condensed matter and cold atomic gases
Beautiful open problems: out of equilibrium, disorder, coupled chains, impurities, etc. ……
Requires interplay of analytical and numerical techniques (and new ideas!) to make progress